Authentic Problems: Expanded Square


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How can we design an expanded square where approximately half the area of the original square is flipped to the outside?

Students design an expanded square that includes symmetrical patterns by reflecting irregular shapes cut from inside a coloured paper square. Their finished design needs to have about half of the coloured paper outside the original square perimeter. Students will explore different methods to determine whether they have removed about half of the original square and adjust their design accordingly. They share their designs describing the symmetrical patterns created and the transformations used. To convince others their design has met the given requirements, students will share the thinking and methods used to create the finished design.

Read the Teachers' Guide before using this resource.


Lesson 1: Discover

Students are introduced to expanded squares (based on the Eastern art of Notan) and explore their characteristics. They construct their first expanded square and discuss its features, including symmetry.

Lesson 2: Devise

Students use fractions to describe the amount of white space in some partially coloured grid squares, and then in some expanded square examples. They consider what fraction of white space is best for visual balance. Students begin to design an expanded square that has approximately half the original square flipped to the outside.

Lesson 3: Develop

Students produce mathematical evidence to convince others that approximately half the area of their original square has been flipped to the outside. They find areas by counting grid squares, or covering with 1 cm cubes, and perhaps rearranging shapes. They seek constructive feedback on the method they used to determine the fraction of the area that has been flipped and on the mathematical evidence they recorded.

Lesson 4: Defend

Students adjust their designs (or describe possible adjustments) and ensure their evidence is detailed and organised clearly so it can be displayed for others to critique. During the feedback session students provide constructive feedback on the strengths and weaknesses of the mathematical evidence and designs. They reflect on feedback given on their display and on their learning throughout the inquiry.

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